Power Set with Union and Subset Relation is Ordered Semigroup

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Theorem

Let $S$ be a set and let $\powerset S$ be its power set.

Let $\struct {\powerset S, \cup, \subseteq}$ be the ordered structure formed from the set union operation and subset relation.


Then $\struct {\powerset S, \cup, \subseteq}$ is an ordered semigroup.


Proof

From Power Set with Union is Commutative Monoid, $\struct {\powerset S, \cup}$ is a fortiori a semigroup.

From Subset Relation is Ordering, $\struct {\powerset S, \subseteq}$ is an ordered set.


It remains to be shown that $\subseteq$ is compatible with $\cup$.


Let $A, B \subseteq S$ be arbitrary such that $A \subseteq B$.

Thus:

\(\ds A\) \(\subseteq\) \(\ds B\)
\(\ds \leadsto \ \ \) \(\ds \forall T \subseteq S: \, \) \(\ds A \cup T\) \(\subseteq\) \(\ds B \cup T\) Set Union Preserves Subsets: Corollary
\(\ds \leadsto \ \ \) \(\ds \forall T \subseteq S: \, \) \(\ds T \cup A\) \(\subseteq\) \(\ds T \cup B\) Union is Commutative

Hence the result.

$\blacksquare$


Sources